The LEDs on the battery indicate the charge remaining. Press the button and you should get 5 LEDs with a full charge. The LEDs stay on in a steady state as long as you hold trhe button

The LEDs can also indicate an internal failure of the battery (Usually 1 3 and 5 will flash) if the LEDs flash, the battery is either critically low or failing. If charging it doesn't change the LED flashing them the battery will need to be replaced

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Checkthe battery charge by pressing the status button or image on the battery near the LED's.Note:
The lights indicate the current amount of charge on the battery. If all the LED's come on when you press the button, the battery is fully charged. If fewer lights are lit, then the battery is partially charged. If any lights flash, or blink, there could be a problem with the battery.

Its an old trick that got my Dell laptop working
Unplug the laptop
remove the battery
press the power button,...nothing will happen,
wait a few seconds
put battery back in laptop
plug it in
hold upside down
press power button until it boots
If that doesn't work it could be a bad/loose/dead bios battery
What is the model of the laptop?

Forgeet the CMOS battery, worthless exercise.
Most likely from what you describe it is a bad motherboard, and on a system that old I'd say replace not repair.
If you want to fix it, mothebroards are $50 to 100 for that model on EBay

Well, when the power supply is plugged to the computer, there should be a light indicating that the battery is charging. One other thing, does it turn off completely (I should have asked this first) or is it just blocking? No messages on screen?

If the light comes on for three seconds and then go out ,generally means a power distrubution problem . remove the optical drive and memory from the system and try powering it on. If u get the same result , its time to take it to service tech , as your motherboard needs to be checked